Method and System for Determining Formation Properties Based on Fracture Treatment

ABSTRACT

A method and system for determining formation properties based on a fracture treatment that may include collecting data from a fracture treatment for a well. A flow regime of the fracture treatment is determined based on the data. Formation properties may be determined based on the flow regime and the data.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of and claims priority to U.S.application Ser. No. 11/031,874, entitled “Method and System forDetermining Formation Properties Based on Fracture Treatment,” filed onJan. 8, 2005, the entire contents of which is incorporated herein byreference for all purposes.

TECHNICAL FIELD

Fracture stimulation of a well, and more particularly to a method andsystem for determining formation properties based on a fracturetreatment.

BACKGROUND

Oil and gas wells produce oil, gas and/or byproducts from undergroundreservoirs. Oil and gas reservoirs are formations of rock containing oiland/or gas. The type and properties of the rock may vary by reservoirand also within reservoirs. For example, the porosity and permeabilityof a reservoir rock may vary from reservoir to reservoir and from wellto well in a reservoir. The porosity is the percentage of core volume,or void space, within the reservoir rock that can contain fluids. Thepermeability is an estimate of the reservoir rock's ability to flow ortransmit fluids.

Oil and gas production from a well may be stimulated by fracture, acidor other production enhancement treatment. In a fracture treatment,fluids are pumped downhole under high pressure to artificially fracturethe reservoir rock in order to increase permeability and production.First, a pad, which is fracture fluids without proppants is pumped downthe well until formation breakdown. Then, the fracturing fluid withproppants is pumped downhole to hold the fractures open after pumpingstops. At the end of the fracture treatment, a clear fluid flush may bepumped down the well to clean the well of proppants.

An initial, or minifracture, test may be performed before a regularfracture stimulation treatment to calculate formation and fractureproperties. Recently, analysis techniques were extended to theafter-closure period. In this analysis, the after-closure data areanalyzed to calculate formation permeability and reservoir pressure.This calculation hypothesizes the existence of either pseudo-radial orlinear flow during the after-closure period.

SUMMARY

A method and system for determining fracture properties are provided. Inaccordance with one embodiment, a method for determining fractureproperties may include collecting data from a fracture treatment for awell. A flow regime from the fracture treatment is determined based onthe data. In a specific embodiment, formation properties may bedetermined based on the flow regime and the data.

Technical advantages of the method and system may include a moregeneralized, simplified, and/or accurate technique for determiningfracture and formation properties from a fracture treatment. Forexample, the flow regime dominating the post-closure period of afracture treatment may be determined based on measured data rather thangeneral assumptions and/or equations. Accordingly, the type of residualfracture may be determined. Reservoir properties such as permeabilityand pressure may then be determined based on the specific flow regime.

Another technical advantage of one or more embodiments may include animproved full, or other subsequent fracture treatment following aminifracture test or other initial fracture treatment. For example,fracture fluids and proppants as well as the duration of pad, proppant,and flush stages may be optimized or otherwise enhanced based onreservoir permeability and pressure determined from an initial fracturetreatment.

Details of the one or more embodiments of the disclosure are set forthin the accompanying drawings in the description below. Other features,objects, and advantages of some of the embodiments will be apparent fromthe description and drawings, and from the claims. Some, all, or none ofthe embodiments may include advantages described herein.

DESCRIPTION OF DRAWINGS

FIG. 1 illustrates one embodiment of a fracture treatment for a well;

FIGS. 2A-2C illustrate exemplary residual fractures and correspondingflow regimes for the fracture treatment of FIG. 1;

FIG. 3 illustrates one embodiment of the fracture control of FIG. 1;

FIGS. 4A-C illustrate exemplary graphs for determining formationproperties from an exemplary fracture treatment having a post-closurepseudo-radial flow regime;

FIGS. 5A-5D illustrate exemplary graphs for determining formationproperties from an exemplary fracture treatment having a post-closurebilinear flow regime of pressure functions for various flow regimes; and

FIG. 6 illustrates a method for determining fracture and formationproperties based on a fracture treatment.

FIG. 7 illustrates a horizontal segment of a wellbore perpendicular toleast stress (σ_(Hmin));

FIG. 8 illustrates a horizontal segment of a wellbore parallel to leaststress (σ_(Hmin));

FIG. 9 illustrates vertical and horizontal segments of a wellbore;

FIG. 10 illustrates a vertical fracture intercepting a horizontal well;

FIG. 11 is a graph showing a comparison of radial/linear flow tobilinear flow;

FIG. 12 is a graph showing an effect of increasing dimensionlesswellbore radius;

FIG. 13 is a graph showing an effect of tail-in with high conductivity;

FIG. 14 is a graph showing an effect of increasing tail-in radius;

FIG. 15 is a graph showing an effect of decreasing conductivity near awellbore;

FIG. 16 is a graph showing an effect of tail-in on well productivity;

FIG. 17 is a graph showing an effect of tail-in radius on wellproductivity;

FIG. 18 is a plan view schematic of a two-dimensional simulator run fora case of two fractures in a horizontal well;

FIG. 19 is a graph showing system flow rate versus number of fractures;

FIG. 20 is a graph showing cumulative production versus time;

FIG. 21 is a graph showing cumulative production versus number offractures;

FIG. 22 is a graph showing cumulative production versus number offractures for the effect of directional horizontal permeabilities; and

FIG. 23 is a graph showing a prediction of total fracture height asinitiated from various depths based on a Δp=250 psi.—

DETAILED DESCRIPTION

FIG. 1 illustrates one embodiment of a fracture treatment 10 for a well12. The well 12 may be an oil and gas well intersecting a reservoir 14.In this embodiment, the reservoir 14 comprises an underground formationof rock containing oil and/or gas. The well 12 may in other embodiments,intersect other suitable types of reservoirs 14.

The fracture treatment 10 may comprise a mini fracture test treatment orother suitable treatment. In the mini fracture test treatmentembodiment, the fracture treatment 10 may be used to determine formationproperties and residual fracture properties before a regular or fullfracture treatment. The formation properties may comprise, for example,reservoir pressure and formation permeability. The formationpermeability is an estimate of the reservoir rock's ability to flow ortransmit fluids. In other embodiments, the fracture treatment 10 maycomprise a regular or full fracture treatment, a follow-on fracturetreatment, a final fracture treatment or other suitable fracturetreatment.

The well 12 may include a well bore 20, casing 22 and well head 24. Thewell bore 20 may be a vertical bore, a horizontal bore, a slanted boreor other deviated bore. The casing 22 may be cemented or otherwisesuitably secured in the well bore 12. Perforations 26 may be formed inthe casing 22 at the level of the reservoir 14 to allow oil, gas, andby-products to flow into the well 12 and be produced to the surface 25.Perforations 26 may be formed using shape charges, a perforating gun orotherwise.

For the fracture treatment 10, a work string 30 may be disposed in thewell bore 20. The work string 30 may be coiled tubing, sectioned pipe orother suitable tubing. A fracturing tool 32 may be coupled to an end ofthe work string 30. The fracturing tool 32 may comprise a SURGIFRAC orCOBRA FRAC tool manufactured by HALLIBURTON or other suitable fracturingtool. Packers 36 may seal an annulus 38 of the well bore 20 above andbelow the reservoir 14. Packers 36 may be mechanical, fluid inflatableor other suitable packers.

One or more pump trucks 40 may be coupled to the work string 30 at thesurface 25. The pump trucks 40 pump fracture fluid 58 down the workstring 30 to perform the fracture treatment 10. The fracture fluid 58may comprise a fluid pad, proppants and/or a flush fluid. The pumptrucks 40 may comprise mobile vehicles, equipment such as skids or othersuitable structures.

One or more instrument trucks 44 may also be provided at the surface 25.The instrument truck 44 includes a fracture control system 46 formonitoring and controlling the fracture treatment 10. The fracturecontrol system 46 communicates with surface and/or subsurfaceinstruments to monitor and control the fracture treatment 10. In oneembodiment, the surface and subsurface instruments may comprise surfacesensors 48, down-hole sensors 50 and pump controls 52.

Surface and down-hole sensors 48 and 50 may comprise pressure, rate,temperature and/or other suitable sensors. Pump controls 52 may comprisecontrols for starting, stopping and/or otherwise controlling pumping aswell as controls for selecting and/or otherwise controlling fluidspumped during the fracture treatment 10. Surface and down-hole sensors48 and 50 as well as pump controls 52 may communicate with the fracturecontrol system 46 over wire-line, wireless or other suitable links. Forexample, surface sensors 48 and pump controls 52 may communicate withthe fracture control system 46 via a wire-line link while down-holesensors 50 communicate wirelessly to a receiver at the surface 25 thatis connected by a wire-line link to the fracture control system 46. Inanother embodiment, the down-hole sensors 50 may upon retrieval from thewell 12 be directly or otherwise connected to fracture control system46.

In operation, the fracturing tool 32 is coupled to the work string 30and positioned in the well 12. The packers 36 are set to isolate thereservoir 14. The pump trucks 40 pump fracture fluid 58 down the workstring 30 to the fracturing tool 32. The fracture fluid 58 exits thefracturing tool 32 and creates a fracture 60 in the reservoir 14. In aparticular embodiment, a fracture fluid 58 may comprise a fluid padpumped down the well 12 until breakdown of the formation in thereservoir 14. Proppants may then be pumped down-hole followed by a clearfluid flush. The fracture treatment 10 may be otherwise suitablyperformed.

FIGS. 2A-C illustrate a plurality of exemplary flow regimes, notnecessarily to scale, formed in the reservoir 14 by the fracturetreatment 10. In particular, FIG. 2A illustrates one embodiment of apseudo-radial flow regime 100. FIG. 2B illustrates one embodiment of abilinear flow regime 110. FIG. 2C illustrates one embodiment of a linearflow regime 120. Other flow regimes may comprise, for example, aspherical flow regime.

Referring to FIG. 2A, the pseudo-radial flow regime 100 comprisesconverging streamlines 102. In this embodiment, iso-potential lines arecircular and extend radially from the well bore 20. The pseudo-radialflow regime 100 may exist, for example, when the period of the fracturetreatment 10 is fairly short and little or no residual fractureconductivity remains after the fracture treatment 10 or afterunrealistically long shut in time.

Referring to FIG. 2B, the bilinear flow regime 110 comprises a first setof parallel stream lines 112 and a second set of parallel stream lines114 perpendicular to the first set of streamlines 112. In thisembodiment, the fracture 60 offers some resistance to fluid flow with asignificant portion of the total pressure drop occurring in the fracture60. Thus, the bilinear flow regime 110 is controlled by the pressuredrop due to linear flow inside the fracture 60 and the pressure drop dueto linear flow in the reservoir 14 surrounding the fracture 60.

Bilinear flow may be present, for example, where the fracture 60 is longor did not completely close, thus maintaining some residualconductivity. The bilinear flow regime 110 may also exist where thefracture treatment 10 comprises an acidized fracture or when release offragments from the reservoir 14 that act as proppants. As anotherexample, if propagation of the fracture 60 has a plastic component, thefracture 60 may maintain some finite width even at closure andaccordingly cause the bilinear flow regime 110.

Referring to FIG. 2C, the linear flow regime 120 comprises parallelstream lines 122. In this embodiment, no appreciable pressure dropoccurs inside the fracture 60. The linear flow regime 120 may exist, forexample, where the fracture 60 stays open with a high dimensionlessconductivity. Dimensionless fracture conductivity may be determined fromthe product of fracture width and fracture permeability divided by theproduct of formation permeability and fracture half length. This mayoccur, for example, where permeability of the reservoir 14 is low andthe fracture treatment 10 is conducted with proppants. It may also occurwhere the fracture 60 stays open for a long period of time.

FIG. 3 illustrates one embodiment of the fracture control system 46. Inthis embodiment, the fracture control system 46 is implemented as anintegrated computer system such as a personal computer, laptop, or otherstand-alone system. In other embodiments, the fracture control system 46may be implemented as a distributed computer system with elements of thefracture control system 46 connected locally and/or remotely by acomputer or other communication network.

The fracture control system 46 may comprise any processors or set ofprocessors that execute instructions and manipulate data to perform theoperations such as, for example, a central processing unit (CPU), ablade, an application specific integrated circuit (ASIC), or afield-programmable gate array (FPGA). Processing may be controlled bylogic which may comprise software and/or hardware instructions. Thesoftware may comprise a computer readable program coded and embedded ona computer readable medium for performing the methods, processes andoperations of the respective engines.

Referring to FIG. 3, the fracture control system 46 includes a datacollection and processing unit 150, a control engine 152, abefore-closure analysis engine 154, an after-closure analysis engine 156and user interface 158. The fracture control system 46 and/or componentsof the fracture control system 46 may comprise additional, different, orother suitable components.

Data collection and processing unit 150 receives and/or communicatessignals to and from surface and down-hole sensors 48 and 50 as well aspump controls 52. The collection and processing unit 150 may correlatereceived signals to a corresponding measured value, filter the data,fill in missing data and/or calculate data derivatives used by one ormore of the control engine 152, before-closure analysis engine 154and/or after-closure analysis engine 156. The data collection processingunit 150 may comprise data input/output (I/O) and a database or otherpersistent or non-persistent storage.

The control engine 152, before-closure analysis engine 154 andafter-closure analysis engine 156 may each be coupled to the datacollection and processing unit 150 and the user interface 158.Accordingly, each may access data collected and/or calculated and eachmay be accessed by an operator or other user via the user interface 158.The user interface 158 may comprise a graphical interface, a text basedinterface or other suitable interface.

The control engine 152 controls the fracture operation 10. In oneembodiment, for example, the control engine 152 may control the pumptrucks 40 and fluid valves to stop and start the fracture operation 10as well as to start and stop the pad phase, proppant phase and/or flushphase of the fracture operation 10.

The before-closure analysis engine 154 analyzes before-closure data todetermine formation properties of the reservoir 14 and of the well 12.In a particular embodiment, the before-closure analysis engine 154 mayprovide G-function analysis and transient analysis. In this embodiment,the G-function analysis may identify the leak-off mechanism and providea definitive indication of the fracture closure stress. The leak-offmechanism may, comprise for example, normal, pressure dependent leak-offfrom open fissures, fracture height recession, fracture tip extension,and changing compliance. The transient analysis may be used to determineformation permeability. In one embodiment, transient analysis assumesthe fracture propagation model and assumes the fracture has the samearea during pumping and closure. The before-closure engine 154 may alsodetermine the fracture 60 closure point. The closure point may bedetermined by using the G-function analysis.

The after-closure analysis engine 156 analyzes after-closure data todetermine formation and residual fracture properties. In one embodiment,the after-closure analysis engine 156 comprises a flow regime engine160, a formation property engine 162 and a fracture planning engine 160.

The flow regime engine 160 determines a flow regime of the fracture 60.In one embodiment, the flow regime engine 160 uses a graphical method todetermine the flow regime based on data measured and collected duringthe fracture treatment 10. The flow regime engine 160 may also orinstead use computational or other suitable methods to determine theflow regime. In a particular embodiment, the after-closure analysisengine 160 may make no assumptions regarding the flow regime dominatingthe reservoir 14 after closure of the fracture 60 or how the fracture 60propagates during the fracture treatment 10.

In the embodiment in which the flow regime engine 160 uses the graphicalmethod to determine the flow regime, the flow regime engine 160 may plotthe derivative of the pressure differential with respect to time p_(fo)versus total time for the fracture treatment 10 on a log-log scale togenerate a derivative graph. The derivative graph is indicative of theflow regime of the fracture 60. In a particular embodiment, thederivative graph may plot log((t_(p)+Δt)∂p_(fo)/∂t) versuslog(t_(p)+Δt), where p_(fo) is pressure during fall-off period (psia), tis total time (hrs.), t_(p) is injection time (hrs.) and Δt is shut intime (hrs.). The plot of the derivative graph will eventually follow astraight line, the slope of which indicates the flow regime.

The flow regime engine 160 may determine the slope of the straight lineand use the slope to determine flow regime. For example, the flow regimeengine 160 may determine that the pseudo-radial flow regime 100dominates the fluid flow behavior after closure of the fracture 60following the fracture treatment 10 if the slope of the straight line is−1, i.e., forms a horizontal line. In another example, the flow regimeengine 160 may determine that the bilinear flow regime 110 dominatesfluid flow behavior after closure of the fracture 60 following thefracture treatment 10 if the slope of the straight line is −0.75. Instill another example, the flow regime engine 160 may determine that thelinear flow regime 120 dominates the fluid flow behavior after closureof the fracture 60 following the fracture treatment 10 if the slope ofthe straight line comprises −0.5.

The flow regime engine 160 may output the determined flow regime to thedata collection and processing unit 150 for storage, to the formationproperty engine 162 for use in determining formation properties and/orto the user interface 158 for review and/or use by the user.

The formation property engine 162 may determine formation properties ofthe reservoir 14 based on the determined flow regime and the datacollected from the fracture treatment 10. In one embodiment, thereservoir property engine 162 may determine the formation properties ofreservoir pressure and formation permeability. In this embodiment, asdescribed in more detail below, the formation property engine 162 mayuse one or more of the following equations or graphs generated from theequations to determine the formation properties for the indicated flowregimes:

$\begin{matrix}{\mspace{79mu} {{{For}\mspace{14mu} {the}\mspace{14mu} {pseudo}\text{-}{radial}\mspace{14mu} {flow}\mspace{14mu} {regime}\mspace{14mu} 100}:}} & \; \\{\mspace{79mu} {{p_{fo} - p_{i}} = {\frac{1694.4\; V\; \mu}{kh}\frac{1}{\left( {t_{p} + {\Delta \; t}} \right)}}}} & \left( {{pseudo}\text{-}{radial}\mspace{14mu} 1} \right) \\{\mspace{79mu} {{\log \left( {p_{fo} - p_{i}} \right)} = {{\log \left( \frac{1694.4\; V\; \mu}{kh} \right)} - {\log \left( {t_{p} + {\Delta \; t}} \right)}}}} & \left( {{pseudo}\text{-}{radial}\mspace{14mu} 2} \right) \\{\mspace{79mu} {{\log \left( {t\frac{\partial p_{fo}}{\partial t}} \right)} = {{\log \left\lbrack \frac{1694.4\; V\; \mu}{kh} \right\rbrack} - {\log \left( {t_{p} + {\Delta \; t}} \right)}}}} & \left( {{pseudo}\text{-}{radial}\mspace{14mu} 3} \right) \\{\mspace{79mu} {{\log \left( {t\frac{\partial p_{fo}}{\partial t}} \right)} = {\log \left\lbrack \frac{1694.4\; V\; \mu}{kh} \right\rbrack}}} & \left( {{pseudo}\text{-}{radial}\mspace{14mu} 4} \right) \\{\mspace{79mu} {{{For}\mspace{14mu} {the}\mspace{14mu} {bilinear}\mspace{14mu} {flow}\mspace{14mu} {regime}\mspace{14mu} 110}:}} & \; \\{{p_{fo} - p_{i}} = {264.6\frac{V}{h}(\mu)^{0.75}\left( \frac{1}{\varphi \; c_{t}k} \right)^{0.25}\frac{1}{\sqrt{k_{f}w_{f}}}\left( \frac{1}{\left( {t_{p} + {\Delta \; t}} \right)} \right)^{0.75}}} & \left( {{bilinear}\mspace{14mu} 1} \right) \\{{\log \left( {p_{fo} - p_{i}} \right)} = {{\log \left( {264.6\frac{V}{h}(\mu)^{0.75}\left( \frac{1}{\varphi \; c_{t}k} \right)^{0.25}\frac{1}{\sqrt{k_{f}w_{f}}}} \right)} - {0.75\; {\log \left( {t_{p} + {\Delta \; t}} \right)}}}} & \left( {{bilinear}\mspace{14mu} 2} \right) \\{\; {{\log \left( {t\frac{\partial p_{fo}}{\partial t}} \right)} = {{\log \left( {198.45\frac{V}{h}(\mu)^{0.75}\left( \frac{1}{\varphi \; c_{t}k} \right)^{0.25}\frac{1}{\sqrt{k_{f}w_{f}}}} \right)} - {0.75\; {\log \left( {t_{p} + {\Delta \; t}} \right)}}}}} & \left( {{bilinear}\mspace{14mu} 3} \right) \\{{\log\limits_{\cdot}\left( {t^{2}\frac{\partial p_{fo}}{\partial t}} \right)} = {{\log \left( {198.45\frac{V}{h}(\mu)^{0.75}\left( \frac{1}{\varphi \; c_{t}k} \right)^{0.25}\frac{1}{\sqrt{k_{f}w_{f}}}} \right)} + {0.25\; {\log \left( {t_{p} + {\Delta \; t}} \right)}}}} & \left( {{bilinear}\mspace{14mu} 4} \right) \\{\mspace{79mu} {{{For}\mspace{14mu} {the}\mspace{14mu} {linear}\mspace{14mu} {flow}\mspace{14mu} {regime}\mspace{14mu} 120}:}} & \; \\{\mspace{79mu} {{p_{fo} - p_{i}} = {31.05\frac{V}{4\; h}\left( \frac{\mu}{\phi \; c_{t}{kL}_{f}^{2}} \right)^{0.5}\left( \frac{1}{t_{p} + {\Delta \; t}} \right)^{0.5}}}} & \left( {{linear}\mspace{14mu} 1} \right) \\{{\log \left( {p_{fo} - p_{i}} \right)} = {{\log \left\lbrack {31.05\frac{V}{4\; h}\left( \frac{\mu}{\phi \; c_{t}{kL}_{f}^{2}} \right)^{0.5}} \right\rbrack} - {0.5\; {\log \left( {t_{p} + {\Delta \; t}} \right)}}}} & \left( {{linear}\mspace{14mu} 2} \right) \\{{\log \left( {t^{2}\frac{\partial p_{fo}}{\partial t}} \right)} = {{\log \left\lbrack {15.525\frac{V}{4\; h}\left( \frac{\mu}{\phi \; c_{t}{kL}_{f}^{2}} \right)^{0.5}} \right\rbrack} - {0.5\; {\log \left( {t_{p} + {\Delta \; t}} \right)}}}} & \left( {{linear}\mspace{14mu} 3} \right) \\{{\log \left( {t^{2}\frac{\partial p_{fo}}{\partial t}} \right)} = {{\log \left\lbrack {15.525\frac{V}{4\; h}\left( \frac{\mu}{\phi \; c_{t}{kL}_{f}^{2}} \right)^{0.5}} \right\rbrack} - {0.5\; {\log \left( {t_{p} + {\Delta \; t}} \right)}}}} & \left( {{linear}\mspace{14mu} 4} \right) \\{\mspace{79mu} {{t = {\frac{60.675\; {\varphi\mu}\; c_{t}L_{f}^{2}}{k}{hr}}}\begin{matrix}{ c_{t}} & {{total}\mspace{14mu} {formation}\mspace{14mu} {compressibility}\; ({psi})} \\{\mspace{85mu} h} & {{net}\mspace{14mu} {pay}\mspace{14mu} {thickness}\; ({ft})} \\{\mspace{85mu} k} & {{Formation}\mspace{14mu} {permeability}\; ({md})} \\{\mspace{85mu} {kf}} & {{Fracture}\mspace{14mu} {conductivity}\; \left( {{md}\text{-}{ft}} \right)} \\{\mspace{79mu} {Lf}} & {{Fracture}\mspace{14mu} {half}\mspace{14mu} {length}\; ({ft})} \\{\mspace{79mu} p_{fo}} & {{Pressure}\mspace{14mu} {during}\mspace{14mu} {fall}\text{-}{off}\mspace{14mu} {period}\; ({psia})} \\{\mspace{79mu} p_{i}} & {{Initial}{\mspace{11mu} \;}{reservoir}{\mspace{11mu} \;}{pressure}\; ({psia})} \\{\mspace{79mu} t} & {{Time}\; ({hrs})} \\{\mspace{79mu} t_{p}} & {{Pumping}\mspace{14mu} {time}\; ({hrs})} \\{\mspace{79mu} V} & {{Injected}{\mspace{11mu} \;}{volume}\mspace{14mu} {into}{\mspace{11mu} \;}{the}\mspace{14mu} {chamber}\; \left( {{bb}\; 1} \right)} \\{\mspace{79mu} {\Delta \; t}} & {{Shut}\text{-}{in}\mspace{14mu} {time}\; ({hrs})} \\{\mspace{79mu} w_{f}} & {{Fracture}\mspace{14mu} {width}} \\{\mspace{79mu} \mu} & {{Viscosity}\; ({cp})} \\{\mspace{79mu} \varphi} & {Porosity}\end{matrix}}} & \left( {{linear}\mspace{14mu} 5} \right)\end{matrix}$

For each flow regime, equation 1 describes the behavior of the pressuredata during the post-closure period of the fracture treatment 10.Equations 2-4 provide specialized log-log and derivative forms ofequation 1. In particular equation 2 is a log of equation 1 whileequations 3-4 are derivatives and logs of equation 1.

For the pseudo-radial flow regime 100, generating the derivative graphusing equation pseudo-radial 2 yields a straight line with a slope of−1. Equation pseudo-radial 3 is independent of initial reservoirpressure, thus its plot is only a function of the observed pressure andtime. Generating the derivative graph using equation pseudo-radial 3also yields a straight line with a slope of −1. Equation pseudo-radial 4is a variation on equation pseudo-radial 3 that may be used for the samepurpose. However, equation pseudo-radial 4 produces a straight line witha slope of 0.

To determine formation properties for the pseudo-radial flow regime 100the formation property engine 162 may plot pressure and time data usingequation pseudo-radial 1 to generate a Cartesian graph of p_(fo)−p_(i)versus 1/(t_(p)+Δt). The formation property engine 162 may determine theintercept from the Cartesian graph which is the reservoir pressure. Withthe reservoir pressure, formation permeability may be determined fromequations pseudo-radial 1-4. In a particular embodiment, the formationproperty engine 162 determines formation permeability using equation togenerate a logarithmic plot of (p_(fo)−p_(i)) versus (t_(p)+Δt). In aspecific embodiment, the intercept of this plot, b_(r), may be used todetermine formation permeability for the pseudo-radial flow regime 100using

$k = {\left( \frac{1694.4V\; \mu}{b_{r}h} \right).}$

In this case, viscosity of the formation fluid is used.

For bilinear flow regime 110, generating the derivative graph usingequation bilinear 2 yields a straight line with a slope of −0.75.Equation bilinear 3 is independent of initial reservoir pressure, thusthe plot is the only function of the observed pressure and time.Generating the derivative graph using equation bilinear 3 also yields astraight line with a slope of −0.75. Equation bilinear 4 is a variationof bilinear equation 3 that may be used for the same purpose. However,equation bilinear 4 produces a straight line with a slope of 0.25.

To determine formation properties for the bilinear flow regime 110, theformation property engine 162 may plot pressure and time data usingequation bilinear 1 to generate a Cartesian graph of p_(fo)−p_(i) versus(1/(t_(p)+Δt)^(0.75). The formation property engine 162 may determinethe intercept from the Cartesian graph which is a reservoir pressure.With a reservoir pressure, formation permeability may be determined fromequations bilinear 1-4. In a particular embodiment, the formationproperty engine 162 may determine formation permeability using equationbilinear 2 to generate a logarithmic plot of (p_(fo)−p_(i)) versus(t_(p)+Δt).

In a specific embodiment, for the bilinear flow regime 110, theintercept, b_(r), is a function of both permeability and fractureconductivity and may be directly used to determine formationpermeability using

$k = {264.6\frac{V}{h}\frac{\mu}{b_{r}}\frac{1}{\left( {2.637\; t_{ef}} \right)^{0.25}}}$

where t_(ef) is the time to end of the bilinear flow. This calculationassumes that fracture length did not change and relies on observance ofthe end of the bilinear flow. If the end of the bilinear flow period isnot observed for the fracture treatment 10, the last point on thestraight line with slope of −0.75 may be used to calculate an upperbound of the formation permeability. For this calculation, viscosity ofthe filtrate fluid that leaked into the formation during theminifracture test may be used as the bilinear flow regime 110 reflectsconditions inside and near the fracture 60.

For the linear flow regime 120, generating the derivative graph usingequation linear 2 yields a straight line with a slope of −0.5. Equationlinear 3 is independent of initial reservoir pressure, thus the plot isonly a function of the observed pressure and, time. Generating thederivative graph using equation linear 3 also yields a straight linewith a slope of −0.5. Equation linear 4 is a variation of equation 3 andmay be used for the same purpose. However, equation linear 4 produces astraight line with a slope of 0.5.

To determine formation properties for the linear flow regime 120, theformation property engine 162 may plot pressure and time data accordingto equation 1 to generate a Cartesian graph. The formation propertyengine 162 may determine the intercept from the Cartesian graph which isthe reservoir pressure. With the reservoir pressure, formationpermeability may be determined from equations 1-4. In a particularembodiment, the formation property engine 162 may determine formationpermeability using equation 2. The end of linear flow (end of −0.5 forequation linear 3 and 0.5 for equation linear 4) occurs at dimensionlesstime of 0.016 and may be calculated using equation linear 5.

The formation property engine 162 may provide the reservoir pressureand/or formation permeability to the data collection and processing unit150 for storage, to the fracture planning engine 162 for planning of asubsequent fracture treatment or to the user interface 158 for reviewand use by the user. The fracture planning engine 164 may modify pumptimes, pump pressures, fracture fluids including the pad, proppants andflush, based on the formation properties. The modification of thesubsequent fracture treatment may include planning of the subsequentfracture treatment based on the formation properties or may comprise anyadjustment to a planned fracture treatment to improve the viability,usefulness, usability, ease of use, efficiency, accuracy, cost or resultof the subsequent fracture treatment.

FIGS. 4A-D illustrate exemplary graphs for the pseudo-radial flow regime100. In particular, FIG. 4A illustrates a treatment graph 200 forfracture treatment 10. FIG. 4B illustrates a derivative graph 210. FIG.4C illustrates a Cartesian graph 220. FIG. 4D illustrates a log-loggraph 230.

Referring to FIG. 4A, the treatment graph 200 plots bottom hole pressure202 and injection rate 204 versus time 206 for the fracture treatment10. Bottom hole pressure 202, injection rate 204 and time 206 may eachbe measured using one or more instruments or determined from orotherwise based on measured parameters. The fracture treatment 10 ofFIG. 4A comprise an average injection rate of 18.8 bpm, with variationsfrom 5 to 25 bpm. The instantaneous shut-in pressure (ISIP) was 13,760psi, which resulted in a 0.97 psi/ft fracture gradient. The G-functionanalysis of the before-closure analysis engine 154 indicates closure at12,578 psi.

Referring to FIG. 4B, the derivative graph 210 plots the derivative ofpressure with respect to time 212 versus total time 214 on a log-logscale. From the derivative graph 210, the flow regime engine 160determines the plot 216 follows a straight line having a slope of −1.From the slope, the flow regime engine 160 determines the after-closureperiod of the fracture treatment 10 is dominated by the pseudo-radialflow regime 100.

Referring to FIG. 4C, the Cartesian graph 220 plots bottom-hole pressure222 versus the time reciprocal 224 to determine the intercept 228 of astraight line 226 at reciprocal time 0 (infinite shut-in time) which isthe reservoir pressure. For exemplary plot 220, reservoir pressure is9,000 psi.

Referring to FIG. 4D, the log-log graph 230 plots the change in pressure232 versus the time reciprocal 234 using the reservoir pressure. Theintercept of the straight line 238 is determined. In the exemplaryembodiment, the intercept is 420 psi which may used to calculate apermeability of 7.12 and where net pay thickness is 20 ft, viscosity is0.0037 cp and injected volume is 288.6 bbl. In one embodiment, heightmay be determined from logs, viscosity is a fluid property, and injectedvolume is known as a test parameter.

FIGS. 5A-D illustrate exemplary graphs for the bilinear flow regime 110.In particular, FIG. 5A illustrates a treatment graph 250 for anotherfracture treatment 10. FIG. 5B illustrates a derivative graph 260. FIG.5C illustrates a Cartesian graph 270. FIG. 5D illustrates log-log graph280.

Referring to FIG. 5A, the treatment graph 250 plots bottom-hole pressure252 and injection rate 254 versus time 256 for the fracture treatment10. Bottom-hole pressure 252, injection rate 254 and time 256 may eachbe measured using one or more instruments or determined or otherwisebased on measured parameters. The fracture treatment 10 of FIG. 5Acomprises a constant injection rate of 10.5 bpm. The instantaneous ISIPwas 13,508 psi, which resulted in a 1.14 psi/ft fracture gradient. The Gfunction analysis of the before-closure analysis engine 154 indicatesclosure at 11,505 psi.

Referring to FIG. 5B, the derivative graph 260 plots the derivative ofpressure with respect to time 262 versus total time 264 on a log-logscale. From the derivative graph 260, the flow regime engine 160determines the plot 266 follows a straight line having a slope of 0.75.From the slope, the flow regime engine 160 determines the after-closureperiod of the fracture treatment 10 is dominated by the bilinear flowregime 110.

Referring to FIG. 5C, the Cartesian graph 270 plots bottom-hole pressure272 versus the time reciprocal 274 to determine the intercept 278 of thestraight line 276 at reciprocal time 0 which is a reservoir pressure.For exemplary plot 276, reservoir pressure is 7,550 psi.

Referring to FIG. 5D, the log-log graph 280 plots a change in pressure282 versus the time reciprocal 284 using the reservoir pressure. Theintercept of the straight line 288 is determined. In the exemplaryembodiment, the intercept is 933 psi which may be used with the finalpoint on the straight line to calculate an upper bound of permeabilityof 0.5763 and where net pay thickness is 27 ft., viscosity is 0.344 cpand injected volume is 189.7 bbl.

As previously described, the linear flow regime 120 may be similarlydetermined. As also previously described, pseudo-radial, bilinear andlinear flow may be determined for horizontal and other wells. Forexample, for a vertical fracture intersecting a horizontal well, theflow regime may be determined as described in detail above if thefracture is longitudinal relative to the horizontal well. If thefracture is transverse with respect to the horizontal well bore, then wemay have, for example the pseudo-radial flow regime 100 if the fracturecloses with little or no fracture length or the shut-in time is verylong; a linear-radial flow regime corresponding to the bilinear flowregime 110 in a vertical well case or the linear flow regime 120 if thedimensionless fracture conductivity is very high, for example. For thelinear-radial flow regime which is described by Soliman, M. Y., Hunt, J.L., and El-Rabaa, A.: “Fracturing Aspects of Horizontal Wells,” JPT,August 1990, the basic equations may be used to develop specializedplots for determining fracture and formation properties. Theabove-referenced article is reproduced below.

FIG. 6 illustrates one embodiment of a method for determining formationproperties based on a fracture treatment 10. In this embodiment,formation properties are determined based on a mini fracture testtreatment. The method begins at step 300 in which the mini fracture testtreatment is performed. As previously described, the mini fracture testtreatment may comprise a pad phase, a proppant phase and a flush phase.Next, at step 302, data is collected from the mini fracture treatment.The data may comprise pressure, rate and time data as well as other dataused in the flow regime equations. For example, the data may includereservoir data such as fluid viscosity, net pay thickness and totalformation height as well as well data such as well bore radius. The datamay be processed by filtering the data, filling in missing data,determining differentials and derivatives and/or storing the data.

At step 304, the flow regime is determined. As previously described, theflow regime may be a pseudo-radial flow regime 100, a bilinear flowregime 110, a linear flow regime 120 or other suitable flow regime. Theflow regime may be determined based on a function of pressure versustime for the mini fracture test treatment. As previously described, inone embodiment, the flow regime may be determined based on thederivative graph plotting the log of pressure with respect to timeversus the log of total time.

The initial reservoir pressure may be determined at step 306. Theinitial reservoir pressure is determined based on the flow regime usingequations, methods, processes and/or data specific to the flow regime.As previously described, the initial reservoir pressure may bedetermined from the intercept of the Cartesian graph for the flowregime.

At step 308, formation permeability is determined. The formationpermeability may be determined based on the reservoir pressure and theflow regime determined. In one embodiment, as previously described, theformation permeability is determined using the intercept of the log-loggraph.

As previously described, these and/or other reservoir properties mayinstead be determined computationally using the indicated equations,derivatives thereof or other suitable equations.

At step 310, a subsequent fracture is modified based on the reservoirproperties determined from the mini fracture test treatment. Aspreviously described, the pump times, pump pressure and/or fracturefluids of the subsequent fracture treatment may be modified based on thereservoir properties. Step 310 leads to the end of the process.

The above-referenced article, “Fracturing Aspects of Horizontal Wells,”is reproduced below.

Summary. This paper discusses the main reservoir engineering andfracture mechanics aspects of fracturing horizontal wells. Specifically,the paper discusses fracture orientation with respect to a horizontalwellbore, locating a horizontal well to optimize fracture height,determining the optimum number of fractures intercepting a horizontalwell. and the mechanism of fluid flow into a fractured horizontal well.

Introduction. Interest in horizontal well drilling and completions hasincreased during the last few years. The significant advances indrilling and monitoring technology have made it possible to drill,guide, and monitor horizontal holes, making horizontal drilling not onlypossible but also consistently successful. Most wells have beencompleted as drainholes. These drainholes have been used in primaryproduction and in EOR.

Papers on drilling, completion, well testing, and increased productionof horizontal vs. vertical wells have been presented in the petroleumliterature.¹⁻¹⁰ Many papers²⁻⁵ have dealt with steady-state productionincrease of horizontal wells over vertical wells. Graphs and equationshave been presented for calculating steady-state ratios for bothfractured and unfractured wells. Ref. 2 provides a recent review of thistechnology. Other authors⁶⁻⁹ have studied the transient behavior ofpressure response during a drawdown or a buildup of a drainhole. Theliterature lacks comprehensive studies on fracturing horizontal wells,and none of the studies cited above discussed this subject. Only Karcheret al.¹⁰ studied production increase caused by multiple fracturesintercepting a horizontal hole. Using a numerical simulator, Karcherstudied steady-state behavior of infinite-conductivity fractures.

Stability of horizontal holes during drilling is another importantaspect of horizontal well technology. It has been found¹¹ that thedegree of stability of horizontal holes depends on the relativemagnitude of the three principal stresses and the orientation of thewellbore with respect to the minimum horizontal stress.

Although productivity of horizontal wells could be two to five timeshigher than productivity of vertical wells, fracturing a horizontal wellmay further enhance its productivity, especially when formationpermeability is low. Presence of shale streaks or low verticalpermeability that impedes fluid flow in the vertical direction couldmake fracturing a horizontal well a necessity.

This paper discusses fracturing horizontal wells from both reservoirengineering and fracture mechanics points of view. Our goal is to shedsome light on important aspects of fracturing horizontal wells.

Stress magnitude and Orientation. The first parameter to be determinedis the fracture orientation with respect to the wellbore. Becausefractures are always perpendicular to the least principal stress, thequestions actually concern wellbore- and stress-orientationmeasurements.

In what direction will induced fractures occur?

What is the anticipated fracture geometry?

What is the optimum length of the perforation interval?

What is the optimum treatment size?

What are the expected fracturing pressures?

Data necessary for planning a fracturing treatment are the mechanicalproperties of the formation, the orientation and magnitude of the leastprincipal stress, the variation in stresses above and below the targetformation, and the leakoff characteristics of the formation.

It is commonly accepted that, at depths usually encountered in the oilfield, the least principal stress is a horizontal stress. It also can beshown that the induced fracture will be oriented perpendicular to theleast principal stress. The result is that a fracture created by atreatment will be in a vertical plane. If the horizontal segment isdrilled in the direction of the least stress, several vertical fracturesmay be spaced along its axis wherever perforations are present. Thisspacing is one of the design parameters to be selected. Usually, this isinvestigated with numerical simulators. If the horizontal segment isdrilled perpendicular to the least stress, one vertical fracture will becreated parallel to the well. FIGS. 1 and 2 show fracture direction vs.well direction.

When the wellbore is not in one of these two major directions, severalscenarios may occur, depending on the angle between the wellbore and thestress direction and one the perforation distribution and density. Inthis paper, only the presence of fractures perpendicular or parallel tothe wellbore is discussed.

Determining Magnitude and Orientation of Least Principal Stress. Iffield history does not clearly reveal the orientation and magnitude ofthe least principal stress, on-site tests should be performed todetermine these parameters. Three methods to determine stress magnitudeand/or orientation exist. Microfracturing, described by Daneshy etal.,¹² may be used to measure the least principal stress and fractureorientation directly. Long-spaced sonic logging may be used to estimatestress magnitude; however, logging has the disadvantage of ignoringtectonic stresses. Strain relaxation may also be used to estimatemagnitude and orientation.¹³ Because the openhole microfracturingtechnique is a direct measurement of stress magnitude and orientation,it is recommended for new reservoirs.

To collect the necessary data, it is recommended that, first, the wellbe drilled vertically through the pay zone and that tests to measurestress magnitude and orientation be performed. Drillstem tests and/orlogging can be performed on the vertical section to determine otherformation parameters.

At the end of these tests, the hole may be plugged back and kickoff canbe performed in the direction determined by the microfracture test. Inthis manner, the most accurate determinations are made in the actualtarget formation, as close as practically possible to the location inwhich fracturing treatments will be performed, without drilling a newvertical well. FIG. 3 shows this procedure.

Fracture Direction With Respect to Wellbore. As mentioned earlier,deciding on fracture orientation with respect to the wellbore isextremely important. One should decide whether designs similar to thoseof FIGS. 1 and 2 should be considered.

If feasible, it is preferable to create effective multiple fracturesbecause of the accelerated production they generate. Here, a simplifiedanalytical model is used to study the effectiveness of fracturesperpendicular to the wellbore. The model considered assumes a wellboreintercepting the fracture plane, as shown in FIG. 4. It assumes thatfluid flows linearly from the formation into the fracture, and thenflows radially inside the fracture into the wellbore. Although thisassumption implies an early-time solution, the results are validqualitatively over a longer time span. In this aspect, the model issimilar to those of Cinco-Ley and Samaniego-V.¹⁴ and Schulte.¹⁵ Toexamine the effect of a tail-in technique on well performance, the modelincorporated a step change in conductivity, similar to the methodpresented in Ref. 16.

In our model, the fracture is assumed to have two distinctconductivities that are radially discontinuous. The governing partialdifferential equation and the final solution for both constant-rate andconstant-pressure cases are presented in Appendices A and B. Thegoverning equations were solved with the Laplace transform. The Laplacetransformation of the solution was reinverted with the Stehfestalgorithm.¹⁷ Various aspects of the solution were studied and arepresented in FIGS. 5 through 11.

FIG. 5 compares the performance of a vertical well that intercepts avertical fracture with the performance of a horizontal well thatintercepts a vertical fracture (perpendicular to well). As shown, forC_(fD)=10, the horizontal well has a significantly higher pressure dropfor the same production rate. The high pressure drop encountered in thecase of a horizontal well intercepting a vertical fracture results fromthe radial flow of fluid inside the fracture as fluid converges towardthe wellbore. C_(fD)≧50 would produce a pressure drop similar to that ofa vertical well that intercepts a vertical fracture with C_(fD)=10,which is evident if FIGS. 5 and 7 and compared.

Effect of Fracture Conductivity. In this section, the fractureorientation with respect to the wellbore is as shown in FIG. 4. Themodel is the same as considered previously.

As FIG. 5 shows, the high pressure drop exhibited by the horizontal wellintercepting a vertical fracture perpendicular to the wellbore is causedby fluid within the fracture converging radially toward the wellbore.This is similar to the pressure drop around an unfractured well. If thewellbore radius increases, the pressure drop necessary to produce a welldecreases (FIG. 6). FIG. 6 also implies that an infinite conductivitymay exist inside the fracture up to the defined r_(wD). For example, thecurve for r_(wD)=0.5 can be explained as a curve for a very small radiusfracture or that for a large-radius fracture with infinite conductivityup to a radius of 0.5x_(f) and C_(fD)=10 from a radius of 0.5x_(f) to aradius x_(f).

The effect of higher-conductivity tail-in is examined further in FIGS. 7and 8. FIG. 7 shows the effect of lower conductivity away from thewellbore. It is assumed that the first half of the fracture ismaintained at C_(fD)=500 and that, for the second half, C_(fD) was 50,20, or 10. In the last three cases, fracture behavior was uniform withC_(fD) 500 until t_(D)=3×10⁻⁷, where the curves exhibited a significantincrease in pressure drop. FIG. 8 examines the effect of tail-in radius,where the radius of higher conductivity (C_(fD)=50) is set at 0.1, 0.3,and 0.5. FIG. 8 shows that deviation from the uniform fracture behaviordepends on the tail-in radius.

FIGS. 5 through 8 show the importance of having high fractureconductivity, or at least high tail-in conductivity. FIG. 9, on theother hand, shows the devastating effect that lower conductivity nearthe wellbore can have. In FIG. 9, only 10% of the fracture dropped fromC_(fD)=50 to C_(fD)=10, 5, or 1. FIG. 9 shows that, even at early time,well behavior was almost totally controlled by the lower-conductivitytail-in. At later times, the high conductivity will have a somewhatminor effect. FIG. 9 indicates that proppant damage near the wellboreshould be avoided and that a tail-in with a strong proppant should bepumped into the fracture whenever feasible.

The parameters used to produce FIGS. 7 and 8 were also used to produceFIGS. 10 and 11. FIGS. 10 and 11, however, are based on constant flowpressure. Conclusions from FIGS. 7 and 8 are confirmed by FIGS. 10 and11.

Previously work¹⁸ showed that high fracture conductivity was necessaryto minimize the cleanup effect following a fracturing treatment. In thecase of a horizontal well intercepting a vertical fracture, cleanupbecomes much more of a problem because of the radial convergence offluid near the wellbore. The presence of higher water saturation nearthe wellbore effectively reduces fracture conductivity near thewellbore, resulting in behavior similar to low-conductivity tail-in.High fracture conductivity, therefore, is extremely important forhorizontal wells.

This discussion agrees with conclusions reached by Soliman,^(16,19) whostated that fracture performance depends on the magnitude and thedistribution of conductivity and does not depend solely on the averageof fracture conductivities, as concluded by Bennett et al.²⁰ Solimanshowed that a conductivity distribution profile exists where a fracturewith declining conductivity performs as well as a uniform fractureconductivity,¹⁹ in spite of the difference in average fractureconductivity.

Determining the Optimum Number of Fractures. To determine the optimumnumber of fractures intercepting the horizontal wellbore that isnecessary to produce the formation, the following assumptions were made.

1. Fractures are identical in physical dimensions (length, height,width, and conductivity).

2. Fractures are vertical and perpendicular to the wellbore axis.

3. Fracture conductivity is sufficiently high to be assumed infinite.

4. Because the horizontal section is assumed cemented, cased, andperforated at the sections where fractures are created, formation fluidcannot flow directly into the wellbore.

The equations governing fluid flow in the formation and fracture can besolved with a single-phase finite-difference simulator. Thesimulator,¹⁹⁻²¹ which solves the governing equations implicitly, wasapplied to an actual field case. Table 1 gives the reservoir properties,and FIG. 12 is a schematic representing one simulator run for the caseof two fractures. FIGS. 13 through 15 show field case results.

TABLE 1 Well and Reservoir Parameters for FIGS. 19 through 22 k, md 0.10φ, % 13 h, ft 272 p_(i), psia 4,000 T_(bh), ° F. 150 p_(wf), psia 50 A,acres 170 S_(w), % 50 x_(f), ft 136 c_(f), md-ft 1,381 w, in. 0.19

In FIG. 13, total flow rate is plotted vs. the number of fractures atvarious times, while FIGS. 14 and 15 show cumulative production vs. timeand number of fractures, respectively.

FIG. 13 shows that, initially, total flow rate increases as the numberof fractures increases. The total flow rate reaches a maximum, and thenit declines. The number of fractures at which the maximum flow rateoccurs declines with time, reaching five fractures after 1 month butdeclining to only two fractures after 24 months. The decline in totalflow rate is caused by reservoir depletion. The optimum number offractures is better determined from FIGS. 14 and 15, which show that,for the case under consideration, five fractures represent the optimumnumber of fractures necessary to produce the reservoir. This number mayvary if economic considerations are included. Note that the optimumnumber of fractures depends on formation and fluid properties.

Reservoir heterogeneity and directional permeability also affect theoptimum number of fractures. The effect of directional permeability onthe optimum number of fractures is investigated with the simulator byvarying the ratio of horizontal permeabilities. FIG. 16 shows theresults of cumulative production vs. the number of fractures at 6 and 24months as a function if directional permeability. The two permeabilitiesconsidered are horizontal, with k_(x) being the permeability parallel tothe fracture plane (perpendicular to the horizontal wellbore axis) andk_(y) being perpendicular to k_(x). The k_(x)/k_(y)=1.0 curves in FIG.16 are the same as those in FIG. 15, and they show five fractures asoptimum. When k_(x)<k_(y) few fractures are needed to produce thereservoir. This is demonstrated for the case with k_(x)/k_(y)=0.10,which results in three fractures being optimum. When k_(y)<k_(x), theoptimum number of fractures increases. In fact, for the case shown inFIG. 16 (k_(x)/k_(y)=10), the optimum number of fracture is >10;economics would dictate this optimum number of fractures.

Consideration of directional permeability simulates the presence oforiented natural fractures. Thus, it is imperative that the presence anddirection of natural fractures be established so that orientation of thehorizontal wellbore and induced fractures with respect to the naturalfractures can be planned to maximize production from the reservoir.

Optimization, in a strict sense, definitely requires the considerationof economic factors, including the cost of a fracturing treatment, theprice of produced hydrocarbon, and the production cost. This strictoptimization may be achieved by studying parameters such as the netpresent value and the benefit/cost ratio.

We use a loose definition of optimization here. The optimum number offractures is the number of fractures at which the rate of increasedproductivity diminishes.

Optimization of Horizontal Well Location. This section presents atechnique to optimize the placement of the horizontal section of a well.The horizontal placement is designed to give optimum fracture height.

A horizontal well location in a given field may be investigated so thatfuture fracturing treatment would expose most of the formation. Analysismay be performed by assuming that vertical fracture growth is controlledby the variation in the closure stress gradient. The main treatmentparameter that could offset this containment criterion is the treatmentpressure. Theoretically, the limits of the treatment pressure to achievecertain fracture growth can be determined. A 2D analysis for fractureextension in a three-layered system was extended for continuous stresschange. By placing the horizontal well along various locations in theformation and by calculating the total fracture height for a certaintreatment pressure, we can determine the optimum location of the well toyield maximum exposure of the pay zone. Derivation of the equation ispresented in Appendix C.

Optimization Example. An actual well was logged between 4,480 and 4,639ft, giving stresses every 10 ft. Generated data are presented in Table2.

TABLE 2 Summary of Stress Data Location Top (ft) Bottom (ft) Stress(psia) Pay Zone 1 3,500 4,480 2,664.1 4,480 4,490 2,601.1 4,490 4,5002,625.1 4,500 4,510 2,144.1 4,510 4,520 2,348.1 4,520 4,530 2.416.14,530 4,540 2,594.1 4,540 4,550 2,273.1 Shale 4,550 4,560 3,070.2 4,5604,570 2,876.2 4,570 4,580 3,330.2 4,580 4,590 3,448.2 4,590 4,6003,244.2 4,600 4,610 3,196.2 Pay Zone 2 4,610 4,620 2,438.2 4,620 4,6302,544.3 4,630 5,000 3,253.3

A simulated fracture is initiated and propagated from the center of eachzone toward the outer layers in 2.5-ft steps. To accomplish thispropagation, the required fracture pressure is calculated with thetechnique described in Appendix C. The total fracture heights (pay-zoneheight plus fracture growth into the upper and lower zones), whichcorrespond to Δp=250 psi [(bottom-hole treatment pressure) minus closurestress] are compared in FIG. 17.

The x axis in FIG. 17 represents the depth of the horizontal well or thedepth of fracture initiation. The y axis gives the amount of fracturepenetration into the upper and lower layers from the horizontal well'scenter (dotted) line. for example, a fracture initiated from a welldrilled at 4,505 ft would penetrate 10 ft above and 30 ft below thewellbore, creating a total height of 40 ft at a pressure of 2,400 psi.As seen from the comparison, a fracture created from a horizontal welldrilled at 4,545 ft would propagate upward in the upper pay zone withvery little penetration in the lower zones. This conclusion would not becritically changed at a reference Δp≠250 psi. The magnitudes of fracturepenetration, however, will be greater; e.g., for Δp=800 psi, a fracturefrom 4,545 ft will penetrate 40 ft, instead of 5 ft, into the lowerlayers and more that 150 ft upward, compared to 70 ft at 250 psi.

Conclusions. If a horizontal well is drilled parallel to the minimumhorizontal stress, multiple fractures may be created. Because of theconvergence of streamlines inside the fracture toward the wellbore, wewould expect to observe a higher pressure drop than is usually observedin a vertical well intercepting a vertical fracture with similarconductivity. Consequently, a very high C_(fD) may be necessary. Atail-in with high conductivity will definitely help to reduce theobserved pressure drop. Fracturing-fluid-cleanup considerations dictatehigh conductivity.

If high or essentially infinite conductivity is feasible, an optimumnumber of fractures may be obtained. This optimum number depends onformation and fluid properties and on the presence of natural fractures.

With the in-situ stress varying through the pay zone, optimum placementof the horizontal wellbore can be determined. Optimum placement is basedon the created fracture's height.

Nomenclature

-   -   a=variable defined by Eq. A-26    -   A=area, acres    -   B=FVF, RB/STB    -   c_(ft)=total fracture compressibility, psi⁻¹    -   c_(t)=total system compressibility, psi⁻¹    -   C=K_(IC)√{square root over (π)}[(1/√{square root over        (h_(t))})−(1/√{square root over (h_(j))})]    -   C_(f)=fracture conductivity, md·ft    -   C_(fD)=dimensionless fracture conductivity    -   C₁, C₂, C₃=variables in Laplace space    -   D=fracture spacing, ft    -   F_(x)=ratio of tail-in length to total fracture length    -   F(y,h_(t))=rock properties function    -   h=formation thickness, ft    -   h_(j)=thickness of jth layer with σ_(j), ft    -   h_(t)=total fracture length, ft    -   I₀=modified Bessel function of first kind, zero order    -   I₁=modified Bessel function of first kind, first order    -   k=formation permeability, md    -   k_(f)=fracture permeability, md    -   k_(x)k_(y)=directional horizontal permeabilities, md    -   K_(I)=stress intensity factor, psi−√{square root over (in.)}    -   K_(IC)=fracture toughness, psi−√{square root over (in.)}    -   K₀=modified. Bessel function of second kind, zero order    -   K₁=modified Bessel function of second kind, first order    -   L=variable defined by Eq. A-27    -   L_(n)=variable defined by Eq. A-24    -   p=formation pressure, psi    -   p_(f)=fracture pressure, psi    -   p_(h)=hydrostatic pressure caused by fluid density, psi    -   p_(i)=initial pressure, psi    -   P_(wD)=dimensionless wellbore pressure    -   P_(wf)=flowing wellbore pressure, psi    -   Δp=pressure difference, psi    -   q=well flow rate, STB/D    -   r=radius, ft    -   r_(w)=wellbore radius, ft    -   s=Laplace transform variable    -   S_(w)=water saturation, percent    -   t=flowing time, hours    -   t_(D)=dimensionless time    -   T_(bh)=bottomhole temperature, ° F.    -   w=fracture width, ft    -   x_(f)=fracture half-length, ft    -   x_(f)=length of tail-in, ft    -   x,y=space coordinates, ft    -   η=formation hydraulic diffusivity, md psi/cp    -   η_(f)=ratio of fracture/formation hydraulic diffusivity    -   μ=fluid viscosity, cp    -   σ_(Hmax)=maximum horizontal stress, psi    -   σ_(Hmin)=minimum horizontal stress, psi    -   σ_(j)=closure stress at initiation zone, psi    -   σ_(n)=closure stress at fracture tip, psi    -   Δσ=σ_(n)−σ_(j), psi    -   Δσ_(j)=difference in stress between adjacent layers, psi    -   φ=formation porosity, fraction    -   φ_(f) fracture porosity, fraction

Subscripts

-   -   D=dimensionless    -   i=initial    -   j=layer    -   t=total    -   1=tail-in    -   2=fracture minus tail-in

Superscript

-   -   ˜=Laplace transform

REFERENCES

-   1. Giger, F. M., Reiss, L. H., and Jourdan, A. P.: “The Reservoir    Engineering Aspects of Horizontal Drilling,” paper SPE 13024    presented at the 1984 SPE Annual Technical Conference and    Exhibition, Houston, September 16-19.-   2. Joshi, S. D.: “A Review of Horizontal Well and Drainhole    Technology,” paper SPE 16868 presented at the 1987 SPE Annual    Technical Conference and Exhibition, Dallas, September 27-30.-   3. Ertekin, T., Sung, W., and Schwerer, F. C.: “Production    Performance Analysis of Horizontal Drainage Wells for the    Degasification of Coal Seams,” JPT (May 1988) 625-32.-   4. Giger, F. M.: “Low-Permeability Reservoir Development Using    Horizontal Wells,” paper SPE 16406 presented at the 1987 SPE/DOE Low    Permeability Reservoirs Symposium, Denver, May 18-19.-   5. Joshi, S. D.: “Augmentation of Well Productivity With Slant and    Horizontal Wells,”: JPT (June 1988) 729-39; Trans., AIME, 285.-   6. Clonts, M. D. and Ramey, H. J. Jr.: “Pressure-Transient Analysis    for Wells With Horizontal Drainholes,” paper SPE 15116 presented at    the 1986 SPE California Regional Meeting, Oakland, April 2-4.-   7. Ozkan, E., Raghavan, R., and Joshi, S. D.: “Horizontal-Well    Pressure Analysis,” SPEFE (December 1989) 567-75; Trans., AIME, 287.-   8. Daviau, F. et al.: “Pressure Analysis for Horizontal Wells,”    SPEFE (December 1988) 716-24.-   9. Goode, P. A. and Thambynayagam, R. K. M.: “Pressure Drawdown and    Buildup Analysis of Horizontal Wells in Anisotropic Media,” SPEFE    (December 1987) 683-97; Trans., AIME, 283.-   10. Karcher, B. J., Giger, F. M., and Combe, J.: “Some Practical    Formulas To Predict Horizontal Well Behavior,” paper SPE 15430    presented at the 1986 SPE Annual Technical Conference and    Exhibition, New Orleans, October 5-8.-   11. Hsiao, C.: “A Study of Horizontal-Wellbore Failure,” SPEPE    (November 1988)489-94.-   12. Daneshy, A. A. et al.: “In-Situ Stress Measurements During    Drilling,” JPT (August 1986) 891-98; Trans., AIME, 281.-   13. El Rabaa, A. W. M. and Meadows, D. L.: “Laboratory and Field    Applications of the Strain Relaxation Method,” paper SPE 15072    presented at the 1986 SPE California Regional Meeting, Oakland,    April 2-4.-   14. Cinco-Ley, H. and Samaniego-V., F.: “Transient Pressure Analysis    for Fractured Wells,” JPT (September 1981) 1749-66.-   15. Schulte, W. M.: “Production From a Fractured Well With Well    Inflow Limited to Part of the Fracture Height,” SPEPE    (September 1986) 333-43.-   16. Soliman, M. Y.: “Design and Analysis of a Fracture With Changing    Conductivity,” J. Cdn. Pet. Tech. (September-October 1986) 62-67).-   17. Stehfest, H.: “Algorithm 368: Numerical Inversion of Laplace    Transforms,” Communications of the ACM, D-5 (January 1970) 13, No.    1, 47-49.-   18. Soliman, M. Y. and Hunt, J. L.: “Effect of Fracturing Fluid and    Its Cleanup on Well Performance,” paper SPE 14514 presented at the    1985 SPE Eastern Regional Meeting, Morgantown, November 6-8.-   19. Soliman, M. Y.: “Fracture Conductivity Distribution Studied,”    Oil & Gas J. (Feb. 10, 1986) 89-93.-   20. Bennett, C. O. et al.: “Influence of Fracture Heterogeneity and    Wing Length on the Response of Vertically Fractured Wells,” SPEJ    (April 1983)219-30.-   21. Soliman, M. Y, Venditto, J. J., and Slusher, G. L.: “Evaluating    Fractured Well Performance by Use of Type Curves,” paper SPE 12598    presented at the 1984 SPE Permian Basin Oil & Gas Recovery    Conference, Midland, March 8-9.-   22. Rice, J. R.: “Mathematical Analysis in the Mechanics of    Fracture,” Treatise on Fracture, Academic Press Inc., New York    City (1962) 2, Chap. 3, 191-276.

Appendix A Solution for Constant-Rate Case

If flow within the fracture is assumed to be radial, the pressurebehavior is described by

$\begin{matrix}{{{{\frac{\partial^{2}p_{{fD}\; 1}}{\partial r_{D}^{2}} + {\frac{1}{r_{D}}\frac{\partial p_{{fD}\; 1}}{\partial r_{D}}} + {\frac{2}{C_{{fD}\; 1}}\frac{\partial p_{D}}{\partial y_{D}}}}}_{y_{D} = 0} = {\frac{1}{\eta}\frac{\partial p_{{fD}\; 1}}{\partial t_{D}}}},{{{where}\mspace{14mu} 0} \leq r_{D} \leq F_{x}},{t_{D} > 0},{and}} & \left( {A\text{-}1} \right) \\{{{{\frac{\partial^{2}p_{{fD}\; 2}}{\partial r_{D}^{2}} + {\frac{1}{r_{D}}\frac{\partial p_{{fD}\; 2}}{\partial r_{D}}} + {\frac{2}{C_{{fD}\; 2}}\frac{\partial p_{D}}{\partial y_{D}}}}}_{y_{D} = 0} = {\frac{1}{\eta}\frac{\partial p_{{fD}\; 2}}{\partial t_{D}}}},} & \left( {A\text{-}2} \right)\end{matrix}$

-   -   where F_(x)≦r_(D)≦∞, t_(D)>0.        -   The initial condition is

p_(fD1)=p_(fD2)=0  (A-3)

Boundary conditions are

$\begin{matrix}{{\frac{\partial^{2}p_{{fD}\; 1}}{\partial r_{D}} = {- \frac{1}{C_{fD}r_{wD}}}},{r_{D} = r_{wD}},} & \left( {A\text{-}4} \right) \\{{{{pfD}\; 1} = p_{f\; D\; 2}},{r_{D} = F_{x}},} & \left( {A\text{-}5} \right) \\{{{C_{{fD}\; 1}\frac{\partial p_{{fD}\; 1}}{\partial r_{D}}} = {C_{{fD}\; 2}\frac{\partial p_{{fD}\; 2}}{\partial r_{D}}}},{r_{D} = F_{x}},{and}} & \left( {A\text{-}6} \right) \\{{\lim\limits_{r_{D}\rightarrow\infty}p_{{fD}\; 2}} = 0.} & \left( {A\text{-}7} \right)\end{matrix}$

Eqs. A-5 and A-6 ensure the continuous change of pressure and rateinside the fracture at the point where the fracture conductivitychanges.

Transient flow in the formation is described by

$\begin{matrix}{{\frac{\partial^{2}p_{D}}{\partial y_{D}^{2}} = \frac{\partial p_{D}}{\partial t_{D}}},{0 < y_{D} < \infty},{t_{D} > 0},} & \left( {A\text{-}8} \right)\end{matrix}$

-   -   with initial conditions of

p_(D)=0, 0<y_(D)<∞, t_(D)=0,  (A-9)

-   -   and boundary conditions

$\begin{matrix}{{p_{D} = p_{fD}},{y_{D} = 0},\; {t_{D} > 0},} & \left( {A\text{-}10} \right) \\{{{\lim\limits_{y_{D}\rightarrow\infty}p_{D}} = 0},{t_{D} > 0},} & \left( {A\text{-}11} \right)\end{matrix}$

-   -   where

p _(fD) =kh(p _(i) −p _(f))/141.2qBμ  (A-12)

and p _(D) =kh(p _(i) −p)/141.2qBμ  (A-13)

Eq. A-10 indicates that formation pressure should be equal to fracturepressure at points of contact. Definitions in Eqs. A-12 and A-13 useformation permeability to achieve the dimensionless form given in Eq.A-1.

t _(D)=0.000264kt/φμc _(t) x _(f) ²,  (A-14)

r _(D) =r/x _(f),  (A-15)

r _(wD) =r _(w) /x _(f),  (A-16)

F _(x) =x _(f) ′/x _(f),  (A-17)

C _(fD) =k _(f) w/kx _(f),  (A-18)

and η_(r) =k _(f) φc _(t) /kφ _(f) c _(f1).  (A-19)

the Three Partial Differential Equations (Esq. A-1, A-2, and A-8) arecoupled with the boundary conditions. By Laplace transform and byalgebraic substitution, the system of equations is reduced to a systemof ordinary differentia equations.

The final expression for the Laplace transform of dimensionless pressureat the wellbore is

{tilde over (p)} _(wD) =C ₁ [C ₂ I ₀(r _(wD) L ₁)+K ₀(r _(wD) L₁)],  (A-20)

$\begin{matrix}{{{{where}\mspace{14mu} C_{1}} = \frac{- 1}{{sC}_{fD}r_{wD}{L_{1}\left\lbrack {{C_{2}{I_{1}\left( {r_{wD}L_{1}} \right)}} - {K_{1}\left( {r_{wD}L_{1}} \right)}} \right\rbrack}}},} & \left( {A\text{-}21} \right) \\{{C_{2} = \frac{{C_{3}{K_{1}\left( {F_{x}L_{1}} \right)}} - {K_{0}\left( {F_{x}L_{1}} \right)}}{{I_{0}\left( {F_{x}L_{1}} \right)} + {C_{3}{I_{1}\left( {F_{x}L_{1}} \right)}}}},} & \left( {A\text{-}22} \right) \\{{C_{3} = {\frac{C_{{fD}\; 1}}{C_{{fD}\; 2}}\frac{L_{1}}{L_{2}}\frac{K_{0}\left( {F_{x}L_{2}} \right)}{K_{1}\left( {F_{x}L_{2}} \right)}}},} & \left( {A\text{-}23} \right)\end{matrix}$

-   -   and

L _(n)=[(2√{square root over (s)}/C _(fDn))+(s/η _(fDn))]^(1/2).  (A-24)

The solution for a uniform fracture may be derived from Eq. A-20 bysetting F_(X)=1.0, Eq. A-25 describes the final solution for auniform-conductivity fracture.

$\begin{matrix}{{\overset{\_}{p}}_{wD} = \frac{K_{0}\left( {r_{wD}L} \right)}{{sC}_{fD}{{LK}_{1}\left( {r_{wD}L} \right)}r_{wD}}} & \left( {A\text{-}25} \right)\end{matrix}$

Eq. A-25 is somewhat different from Schulte's¹⁵ Eq. A-8, which has atypographical error. It should have the √{square root over (a)} term inits denominator. Also, Schulte's equation was for a half circle;consequently, the calculated pressure drop from his equation would betwice the pressure drop calculated with the equation for a full circleat the same production rate. Schulte also ignored the storativity of thefracture; consequently, ∀ in his paper is defined as

a=2√s/C _(fD).  (A-26)

In this paper, L2, which is equivalent to ∀, is defined as

L ²=(2√{square root over (s)}/C _(fD))+(s/η _(fD)).  (A-27)

The solution developed here is basically for early-time producing aradial/linear flow regime comparable to the bilinear flow regime invertical wells.¹⁴ Consequently, only formation permeabilityperpendicular to fracture affects fluid flow. After a long producingtime, permeability both parallel and perpendicular to fracture willaffect fluid flow.

Appendix B Solution of Constant-Pressure Case

The procedure presented above can also be used to solve equations forflow under constant wellbore pressure. Slight modifications, however,are required. First, the dimensionless pressure is defined as

P _(D)=(P _(i) −p)/(P _(i) −P _(wf))  (B-1)

and P _(fD)=(P _(i) −P _(f))/(P _(i) −P _(wf))  (B-2)

The boundary condition at the wellbore is replaced by

P_(fD)=1.0,  (B-3)

-   -   and the dimensionless flow rate is

$\begin{matrix}{q_{D} = {{- C_{fD}}\frac{\partial p_{fD}}{\partial r_{D}}{r_{wD}.}}} & \left( {B\text{-}4} \right)\end{matrix}$

With these definitions, the governing partial differential equations canbe solved for a changing-conductivity fracture, as presented in Eq. B-5.

{tilde over (q)} _(D) =−C _(fD1) C ₁ L ₁ [C ₂ I ₁(r _(wD) L ₁)−K ₁(r_(wD) L ₁)]r _(wD),  (B-5)

$\begin{matrix}{{{{where}\mspace{14mu} C_{1}} = {1/\left\{ {s\left\lbrack {{C_{2}{I_{0}\left( L_{1} \right)}} + {K_{0}\left( L_{1} \right)}} \right\rbrack} \right\}}},} & \left( {B\text{-}6} \right) \\{{C_{2} = \frac{{C_{3}{K_{1}\left( {F_{x}L_{1}} \right)}} - {K_{0}\left( {F_{x}L_{1}} \right)}}{{I_{0}\left( {F_{x}L_{1}} \right)} + {C_{3}{I_{1}\left( {F_{x}L_{1}} \right)}}}},{and}} & \left( {B\text{-}7} \right) \\{C_{3} = {\frac{C_{{fD}\; 1}}{C_{{fD}\; 2}}\frac{L_{1}}{L_{2}}{\frac{K_{0}\left( {F_{x}l_{2}} \right)}{K_{1}\left( {F_{x}L_{2}} \right)}.}}} & \left( {B\text{-}8} \right)\end{matrix}$

The solution for a uniform fracture can be obtained from Eqs. B-5through B-8 by setting F_(X)=1, which yields the following final form:

$\begin{matrix}{{\overset{\_}{q}}_{D} = {\frac{C_{fD}{{LK}_{1}\left( {r_{wD}L} \right)}}{{sK}_{0}\left( {r_{wD}L} \right)}{r_{wD}.}}} & \left( {B\text{-}9} \right)\end{matrix}$

Appendix C Calculating Fracture Pressure for Propagation

The relationship between the total fracture height, the surroundingstresses, and the pressure inside the fracture is found by manipulatingRice's equation²²:

$\begin{matrix}{{K_{1} = {C{\int_{- h_{t}}^{h_{t}}{{p(y)}{F\left( {y,h_{t}} \right)}\ {y}}}}},} & \left( {C\text{-}1} \right)\end{matrix}$

where K₁ is the stress intensity factor at the tips of a fracture,h_(t), loaded by inside pressure p(y). For more than three layers, thesolution can be written in the form:

$\begin{matrix}{{\Delta \; p} = {C + {\Delta\sigma} - {{1/\pi}{\sum\limits_{j = t}^{n}\; {{\Delta\sigma}_{j} \times {\sin^{- 1}\left( {h_{j}/h_{t}} \right)}}}} + {p_{h}.}}} & \left( {C\text{-}2} \right)\end{matrix}$

Although this disclosure has been described in terms of certainembodiments and generally associated methods, alterations andpermutations of these embodiments and methods will be apparent to thoseskilled in the art. Accordingly, the above description of exampleembodiments does not define or constrain this disclosure. Other changes,substitutions, and alterations are also possible without departing fromthe spirit and scope of this disclosure.

1.-28. (canceled)
 29. A method comprising: collecting data from afracture treatment, the fracture treatment fracturing a reservoir byinjecting fluid into the reservoir through a well; determining a flowregime from the fracture treatment by analyzing the collected data for apresence of a plurality of different flow regimes; and performing asubsequent fracture treatment, the subsequent fracture treatment basedon properties determined using the flow regime.
 30. The method of claim29, wherein the data comprises pressure data from the fracturetreatment.
 31. The method of claim 29, wherein the data comprisespost-closure data from the fracture treatment.
 32. (canceled)
 33. Themethod of claim 29, wherein the plurality of different flow regimescomprise at least one of a pseudo-radial flow regime, a bilinear flowregime, or a linear flow regime.
 34. The method of claim 29, wherein thedata comprises pressure data and determining the flow regime from thefracture treatment comprises: plotting pressure data during apost-closure period of the fracture treatment to generate a plot; anddetermining the flow regime from the plot.
 35. The method of claim 34,wherein determining the flow regime comprises determining the flowregime from a slope of the plot.
 36. The method of claim 35, wherein thepressure data is plotted on logarithmic-logarithmic graph.
 37. Themethod of claim 36, wherein the logarithmic-logarithmic graph comprisesa plot of a derivative of a pressure with respect to a time versus atotal time of the fracture treatment.
 38. The method of claim 29,further comprising determining formation properties based on the flowregime and the data.
 39. A method for enhancing a fracture treatment,comprising: performing an initial fracture treatment, the fracturetreatment fracturing a reservoir by injecting fluid into the reservoirthrough a well; collecting data from the fracture treatment; determininga flow regime from the fracture treatment by analyzing the collecteddata for a presence of a plurality of different flow regimes;determining formation properties for the well based on the flow regime;designing a subsequent fracture treatment for the well based on theformation properties; and performing the subsequent fracture treatment.40. The method of claim 39, wherein the plurality of different flowregimes comprise at least one of a pseudo-radial flow regime, a bilinearflow regime or a linear flow regime.
 41. The method of claim 39, whereinthe data comprises pressure data and determine the flow regime from thefracture treatment comprises: plotting pressure data during apost-closure period of the fracture treatment to generate a plot; anddetermining the flow regime from the plot.
 42. The method of claim 41,wherein determining the flow regime comprises determining the flowregime from a slope of the plot.
 43. The method of claim 42, wherein thepressure data is plotted on a logarithmic-logarithmic graph.
 44. Themethod of claim 43, wherein the logarithmic-logarithmic graph comprisesa plot of a derivative of a pressure with respect to a time versus atotal time of the fracture treatment.
 45. An article of manufacturecomprising machine-readable media storing instructions for causing oneor more processors to: collect post-closure data from a fracturetreatment, the fracture treatment fracturing a reservoir by injectingfluid into the reservoir through a well; determine a flow regime fromthe fracture treatment by analyzing the collected post-closure data fora presence of a plurality of different flow regimes; and output the flowregime.
 46. The article of manufacture of claim 45, further comprisinginstructions for causing the one or more processors to plot thepost-closure data on a logarithmic-logarithmic graph and to determinethe flow regime based on plot.
 47. The article of manufacture of claim45, wherein the data comprises post-closure data.
 48. The article ofmanufacture of claim 47, wherein the plurality of different flow regimescomprise at least one of a pseudo-radial flow regime, a bilinear flowregime or a linear flow regime.
 49. The article of manufacture of claim45, wherein the flow regime for the fracture is determined based on aslope of the plot.
 50. The article of manufacture of claim 45, whereinthe logarithmic-logarithmic graph comprises a plot of a derivative of apressure with respect to a time versus a total time of the fracturetreatment.
 51. The article of manufacture of claim 45, furthercomprising instructions for causing the one or more processors todetermine formation properties based on the flow regime and the data.52. A system comprising: data collected from a fracture treatment, thefracture treatment fracturing a reservoir by injecting fluid into thereservoir through a well; means for determining a flow regime from thefracture treatment by analyzing the collected data for a presence of aplurality of different flow regimes; and means for outputting the flowregime.
 53. The system of claim 52, wherein the plurality of differentflow regimes comprise at least one of a pseudo-radial flow regime, abilinear flow regime or a linear flow regime.
 54. The system of claim52, further comprising means for determining formation properties basedon the flow regime and the data.
 55. A method for determining propertiesof a formation, comprising: performing a fracture treatment, thefracture treatment fracturing a formation by injecting fluid into theformation through a well; collecting post-closure pressure data from thefracture treatment; plotting a derivative of the post-closure pressuredata on a logarithmic-logarithmic graph; determining a slope of theplot; determining a presence of one of a pseudo-radial or bilinear flowregime of the fracture treatment based on the slope of the plot;determining formation properties based on the flow regime; and providingthe formation properties to at least one of a data collection unit or aperson.
 56. The method of claim 55, wherein the flow regime comprisesone of pseudo-radial flow regime, a bilinear flow regime and a linearflow regime.
 57. The method of claim 33, wherein the pseudo-radial flowregime comprises converging streamlines, the linear flow regimecomprises parallel stream lines, and the bi-linear flow regime comprisesa first set of parallel stream lines and a second set of parallel streamlines perpendicular to the first set.
 58. The method of claim 38,wherein determining the formation properties comprises determining atleast one of a formation permeability k, an initial reservoir pressurep_(i), a fracture conductivity k_(f), or a fracture half length L_(f).59. The method of claim 38, wherein determining the formation propertiescomprises determining at least one of the formation properties based atleast in part on a sum of a pumping time t_(p) and a shut-in time Δt.60. The method of claim 38, wherein determining the formation propertiescomprises determining the formation properties based at least in part onan intercept (b_(r)) in a logarithmic plot of a pressure versus a time.61. The method of claim 38, wherein determining the formation propertiescomprises determining a formation permeability k based at least in parton parameters including an injected volume V, a viscosity μ, anintercept b_(r) in a logarithmic plot of a pressure versus a time, andnet pay thickness h.
 62. The method of claim 38, wherein determining theformation properties comprises: selecting one or more equations that areappropriate for the flow regime; and using the data and the one or moreequations to determine the formation properties.
 63. The method of claim29, wherein determining a flow regime from the fracture treatment byanalyzing the collected data for a presence of a plurality of differentflow regimes comprises: determining a first value of a slope in a plotrepresenting the collected data, wherein the plurality of different flowregimes correspond to a plurality of different values of the slope; anddetermining the flow regime from the first value of the slope.
 64. Themethod of claim 34, wherein the derivative of the pressure with respectto the time comprises $t\frac{\partial p_{fo}}{\partial t}$ .